Properties

Label 2-74-37.16-c7-0-5
Degree $2$
Conductor $74$
Sign $-0.819 - 0.573i$
Analytic cond. $23.1164$
Root an. cond. $4.80796$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.51 + 2.73i)2-s + (28.9 − 10.5i)3-s + (49.0 + 41.1i)4-s + (−48.7 + 276. i)5-s + 246.·6-s + (−219. + 1.24e3i)7-s + (256. + 443. i)8-s + (−947. + 795. i)9-s + (−1.12e3 + 1.94e3i)10-s + (−3.01e3 − 5.22e3i)11-s + (1.85e3 + 674. i)12-s + (−7.24e3 − 6.07e3i)13-s + (−5.05e3 + 8.75e3i)14-s + (1.50e3 + 8.52e3i)15-s + (711. + 4.03e3i)16-s + (−210. + 176. i)17-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (0.619 − 0.225i)3-s + (0.383 + 0.321i)4-s + (−0.174 + 0.989i)5-s + 0.465·6-s + (−0.241 + 1.37i)7-s + (0.176 + 0.306i)8-s + (−0.433 + 0.363i)9-s + (−0.355 + 0.615i)10-s + (−0.683 − 1.18i)11-s + (0.309 + 0.112i)12-s + (−0.914 − 0.766i)13-s + (−0.492 + 0.852i)14-s + (0.114 + 0.651i)15-s + (0.0434 + 0.246i)16-s + (−0.0103 + 0.00870i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $-0.819 - 0.573i$
Analytic conductor: \(23.1164\)
Root analytic conductor: \(4.80796\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :7/2),\ -0.819 - 0.573i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.643199 + 2.04169i\)
\(L(\frac12)\) \(\approx\) \(0.643199 + 2.04169i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-7.51 - 2.73i)T \)
37 \( 1 + (-6.25e4 - 3.01e5i)T \)
good3 \( 1 + (-28.9 + 10.5i)T + (1.67e3 - 1.40e3i)T^{2} \)
5 \( 1 + (48.7 - 276. i)T + (-7.34e4 - 2.67e4i)T^{2} \)
7 \( 1 + (219. - 1.24e3i)T + (-7.73e5 - 2.81e5i)T^{2} \)
11 \( 1 + (3.01e3 + 5.22e3i)T + (-9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 + (7.24e3 + 6.07e3i)T + (1.08e7 + 6.17e7i)T^{2} \)
17 \( 1 + (210. - 176. i)T + (7.12e7 - 4.04e8i)T^{2} \)
19 \( 1 + (1.87e4 - 6.82e3i)T + (6.84e8 - 5.74e8i)T^{2} \)
23 \( 1 + (5.09e3 - 8.82e3i)T + (-1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 + (-6.21e4 - 1.07e5i)T + (-8.62e9 + 1.49e10i)T^{2} \)
31 \( 1 - 2.52e5T + 2.75e10T^{2} \)
41 \( 1 + (-2.71e4 - 2.28e4i)T + (3.38e10 + 1.91e11i)T^{2} \)
43 \( 1 - 1.91e5T + 2.71e11T^{2} \)
47 \( 1 + (3.11e5 - 5.40e5i)T + (-2.53e11 - 4.38e11i)T^{2} \)
53 \( 1 + (-6.97e4 - 3.95e5i)T + (-1.10e12 + 4.01e11i)T^{2} \)
59 \( 1 + (5.55e4 + 3.15e5i)T + (-2.33e12 + 8.51e11i)T^{2} \)
61 \( 1 + (-1.57e6 - 1.31e6i)T + (5.45e11 + 3.09e12i)T^{2} \)
67 \( 1 + (3.54e5 - 2.00e6i)T + (-5.69e12 - 2.07e12i)T^{2} \)
71 \( 1 + (1.13e6 - 4.13e5i)T + (6.96e12 - 5.84e12i)T^{2} \)
73 \( 1 - 7.81e4T + 1.10e13T^{2} \)
79 \( 1 + (-5.40e5 + 3.06e6i)T + (-1.80e13 - 6.56e12i)T^{2} \)
83 \( 1 + (-6.99e6 + 5.86e6i)T + (4.71e12 - 2.67e13i)T^{2} \)
89 \( 1 + (-1.44e6 - 8.19e6i)T + (-4.15e13 + 1.51e13i)T^{2} \)
97 \( 1 + (5.47e6 - 9.47e6i)T + (-4.03e13 - 6.99e13i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.70440765399936530678362079713, −12.65092677452223669752522962480, −11.52450671829523774937836840176, −10.40766918204209383892463113999, −8.655587623139364286921253295235, −7.81077694512016852692988918730, −6.31469848878554150475007376231, −5.26142299351763540120532101247, −2.96993060421344279609906898525, −2.66556439689707884125834754664, 0.50260738162946491454302416117, 2.36005804166794176646753151997, 4.06824325209370860356476096366, 4.78748644158938145720774538116, 6.74648601325724013530154458415, 7.994696613079853464152768149864, 9.444498483197817081348599131030, 10.33632846482851448589544533417, 11.88222647393360601950381800350, 12.78824459021682165343058731033

Graph of the $Z$-function along the critical line